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A barnstar for your extensive contributions to articles related to statistical mechanics! --HappyCamper

Subjects I'm working on- Wikipedia:Writing better articles

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• References {{note_label|Wood1992||}} {{ref_harvard|Wood1992|Wood, 1992|}}
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• References (Harvard with pages)
<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>
==References==
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=== Bibliography ===
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*{{Cite book|etc |ref=harv}}
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Heavy tail distributions

Heavy tail distributions

Statistical Mechanics

A Table of Statistical Mechanics Articles
Maxwell Boltzmann Bose-Einstein Fermi-Dirac
Particle Boson Fermion
Statistics
Statistics Bose-Einstein statistics Fermi-Dirac statistics
Thomas-Fermi
approximation
gas in a box
gas in a harmonic trap
Gas Ideal gas
Chemical
Equilibrium
Classical Chemical equilibrium

Others:

Work pages

To fix:

 (subtract mean) (no subtract mean) Covariance Correlation Cross covariance Cross correlation see ext Autocovariance Autocorrelation Covariance matrix Correlation matrix Estimation of covariance matrices

Proof: Introduce an additional heat reservoir at an arbitrary temperature T0, as well as N cycles with the following property: the j-th such cycle operates between the T0 reservoir and the Tj reservoir, transferring energy dQj to the latter. From the above definition of temperature, the energy extracted from the T0 reservoir by the j-th cycle is

${\displaystyle dQ_{0,j}=T_{0}{\frac {dQ_{j}}{T_{j}}}\,\!}$

Now consider one cycle of the heat engine, accompanied by one cycle of each of the smaller cycles. At the end of this process, each of the N reservoirs have zero net energy loss (since the energy extracted by the engine is replaced by the smaller cycles), and the heat engine has done an amount of work equal to the energy extracted from the T0 reservoir,

${\displaystyle W=\sum _{j=1}^{N}dQ_{0,j}=T_{0}\sum _{j=1}^{N}{\frac {dQ_{j}}{T_{j}}}\,\!}$

If this quantity is positive, this process would be a perpetual motion machine of the second kind, which is impossible. Thus,

${\displaystyle \sum _{i=1}^{N}{\frac {dQ_{i}}{T_{i}}}\leq 0\,\!}$

Now repeat the above argument for the reverse cycle. The result is

${\displaystyle \sum _{i=1}^{N}{\frac {dQ_{i}}{T_{i}}}=0\,\!}$ (reversible cycles)

In mathematics, it is often desireable to express a functional relationship ${\displaystyle f(x)\,}$ as a different function, whose argument is the derivative of f , rather than x . If we let y=df/dx  be the argument of this new function, then this new function is written ${\displaystyle f^{\star }(y)\,}$ and is called the Legendre transform of the original function.

References

1. {{#invoke:Citation/CS1|citation |CitationClass=journal }} A necessary condition for Planck's law to hold is that the photon number is not conserved, implying that the chemical potential of the photons is zero. While this may be unavoidably true on very long timescales, there are many practical cases that are dealt with by assuming a nonzero chemical potential, which yields an equilibrium distribution which is not Planckian.